Which artificial neural network can mimic biological neurons the most?
Only a small portion of the habituation, sensitization, and classical conditioning behavior of neurons has been primitively simulated in ANN systems. Simulation of actin cytoskeletal machinery1 and other agents of neural plasticity, central to learning new domains, is in its beginnings2.  As of this writing, the complexity of neuron activation dwarfs the models being used in working commercial ANN systems, but the research continues along multiple fronts.


The neuroscience of learning3,
Parallel hardware approaches that better support ANN simulation accuracy4, 5, and
Dynamic frameworks6


This list and the examples referenced in the superscripts, with links below, represent a tiny sample of the information available and the work in progress.

References

[1] Molecular Cell Biology. 4th edition.; Lodish H, Berk A, Zipursky SL, et al.; New York: W. H. Freeman; 2000.; Section 18.1 The Actin Cytoskeleton

[2] NEURON Software; Yale U

[3] Molecular Cell Biology. 4th edition.; Lodish H, Berk A, Zipursky SL, et al.; New York: W. H. Freeman; 2000.; Section 21.7 Learning and Memory

[4] Artificial Neural Networks on Massively Parallel Computer Hardware; Udo Seiffert; University of Magdeburg, Germany

[5] NeuroGrid; Stanford U

[6] Explanation of Dynamic Computational Graph frameworks